Kozma, 29, was designated for assignment by the Yankees Friday after having just nine at-bats in 11 games.
He spent four seasons playing for St. Louis, with his career year coming in 2013 when he played in 143 games and had a homer and 35 RBIs. The Yankees signed him following the '15 season and he spent all of '16 in Triple-A.
Kozma is primarily a shortstop but he can play all four infield positions. He started three games at shortstop for the Yankees this season and also played one game at second base.
Profar made the Opening Day roster for Texas for the first time in his career after a big spring that included a .464 average for Team Netherlands in the World Baseball Classic.
He'll get a chance to play every day for Round Rock.
"I felt the way the rotation was in left field and in the infield, the lack of regular playing time this early in the season, his time was better spent in Triple-A, playing every day and let him get it going," Texas general manager Jon Daniels said. "That's really all it comes down to. It's just a slump. It happens to everybody. He got off to a little bit of a slow start."